Locally minimax efficiency of nonparametric estimates of square-integrable densities

Lithuanian Mathematical Journal -     

Asymptotic properties of nonparametric curve estimates

We consider a class of nonparametric estimators for the regression functionm(t) in the model:y _i =m(t _i ) + _i , 1 i n, t _i [0, 1], which are linear in the observationsy _i . Several limit theorems concerning local and global stochastic and a.s. convergence and limit distributions are given.     

Asymptotic properties of some functions of nonparametric estimates of a density function

Let be a nonparametric estimate of a two-dimensional density f(x,y) constructed with the help of two-dimensional “window.” The main purpose of the paper is to study the asymptotic properties of the marginal moments estimates and differentiable functions , of these moments.     

Stability estimates and structural spectral properties of saddle point problems

For a general class of saddle point problems sharp estimates for Babu?ka’s inf-sup stability constants are derived in terms of the constants in Brezzi’s theory. In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov subspace methods. The theoretical results are applied to two model problems from optimal control with time-periodic state equations. Numerical experiments with the preconditioned minimal residual method are reported.     

The asymptotic properties of the multichannel autoregressive spectral estimates

If we fit a-vector stationary time series using observationsx(1), ...,x(T) with AR models , then the spectral densityf() of {x(t)} can be estimated byf _k ^(T) ()=(2)^– A _k ^(T) (e ^–)^–1 _k ^(T) A _k ^(T) (e ^–i)^–, where are estimates of the variance matrix of(t), the residuals of the best linear prediction. By extending some results for the scalar case, this paper treats the asymptotic properties of the estimates in the multichannel case.     

Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences

Consider the nonparametric regression model Y_ni=g(x_ni)+ε_ni for i=1,…,n, where g is unknown, x_ni are fixed design points, and ε_ni are negatively associated random errors. Nonparametric estimator g_n(x) of g(x) will be introduced and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g_n(x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors (e.g. see J. Multivariate Anal. 25(1) (1988) 100).     

The standard error of nonparametric density estimates

We study the asymptotic behavior of the errors in the Parzen estimators of normal and Cauchy distribution densities for different kernels.Translated from Statisticheskie Metody, pp. 3–18, 1980.     

Strong uniform consistency of nonparametric regression function estimates

Let (X, Y) be a ^dx-valued random vector and let r(t)=E(Y/X=t) be the regression function of Y on X that has to be estimated from a sample (X _i, Y_i), i=1,..., n. We establish conditions ensuring that an estimate of the form

Asymptotic normality of spectral estimates

The asymptotic normality of some spectral estimates, including a functional central limit theorem for an estimate of the spectral distribution function, is proved for fourth-order stationary processes. In contrast to known results it is not assumed that all moments exist or that the process is linear. The data are allowed to be tapered. Using some recent results on the central limit theorem for stationary processes, corollaries are obtained for strong and φ-mixing sequences and linear transformations of martingale differences.     

On a class of Bayesian nonparametric estimates: II. Hazard rate estimates

The Bayes estimation of hazard rates for a family of multiplicative point processes is considered. We study the model for which a hazard rate can be linearly parametrized by a freely varied measure. The weighted gamma process is assumed to be the prior distribution of this measure; the posterior distributions and the posterior means are given in explicit forms. Examples of the evaluation of posterior means are given.The research of this author is supported in part by NSF Grant MCS 81-02523-01.     

Plancherel-type estimates and sharp spectral multipliers

We study general spectral multiplier theorems for self-adjoint positive definite operators on L²(X,μ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R³ and new spectral multiplier theorems for the Laguerre and Hermite expansions.     

Asymptotic theory of nonparametric regression estimates with censored data

For regression analysis, some useful information may have been lost when the responses are right censored. To estimate nonparametric functions, several estimates based on censored data have been proposed and their consistency and convergence rates have been studied in literature, but the optimal rates of global convergence have not been obtained yet. Because of the possible information loss, one may think that it is impossible for an estimate based on censored data to achieve the optimal rates of global convergence for nonparametric regression, which were established by Stone based on complete data. This paper constructs a regression spline estimate of a general nonparametric regression function based on right_censored response data, and proves, under some regularity conditions, that this estimate achieves the optimal rates of global convergence for nonparametric regression. Since the parameters for the nonparametric regression estimate have to be chosen based on a data driven criterion, we also obtain the asymptotic optimality of AIC, AICC, GCV, Cp and FPE criteria in the process of selecting the parameters.     

Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

Maximum entropy interpretation of autoregressive spectral densities

A new proof is given of the maximum entropy characterization of autoregressive spectral densities as models for the spectral density of a stationary time series. The new proof is presented in parallel with a proof of the maximum entropy characterization of exponential models for probability densities. Concepts of entropy, cross-entropy and information divergence are defined for probability densities and for spectral densities.     

Upper estimates of transition densities for stable-dominated semigroups

We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Lévy process.